Differential Equation In Exercises 51-56, solve the differential equation. (d y)/(d x)=(e^x+5)^2

Name: Problem 51, Chapter 8: Calculus of a Single Variable
Uploaded: 2020-01-18T03:34:06-08:00
Duration: 2 min 20 s
Channel: Pravakar Paul
Description: Differential Equation In Exercises 51-56, solve the differential equation. (d y)/(d x)=(e^x+5)^2

step 1 In this problem, we have to solve the differential equation, D -Y -T -X is equal to E to the pow

Differential Equation In Exercises 51-56, solve the...

Key Concepts

General Solution of Differential Equations

The general solution of a differential equation represents the entire family of functions that satisfy the differential equation. It typically includes an arbitrary constant which allows the solution to be tailored to specific initial conditions. This concept is fundamental in understanding how differential equations model systems with a variety of possible states.

Indefinite Integration

Indefinite integration is the process of finding the antiderivative of a function. In the context of solving differential equations, it involves integrating the derivative of the unknown function with respect to the independent variable, resulting in an expression for the original function plus an arbitrary constant. This technique is essential for recovering the function from its rate of change.

Integration of Exponential Functions

When integrating functions that include exponential terms, the properties of the exponential function are used to simplify the integration process. Recognizing standard forms and applying basic integration rules can efficiently lead to the antiderivative. This is particularly useful in problems where the differential equation comprises functions that can be expressed or expanded into forms involving exponentials.

Constant of Integration

The constant of integration arises in the process of indefinite integration, reflecting the fact that the derivative of a constant is zero. It accounts for the family of solutions in a general solution, indicating that without additional initial conditions, the solution to the differential equation is not unique. This concept is critical in distinguishing between a particular solution and a general solution.

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